Abstract.Let k > 1 be an odd integer, t = L 4 J > and q be a prime power. We construct a bipartite, ^-regular, edge-transitive graph CD(k , q) of order v < 2qk~'+l and girth g > k + 5 . If e is the the number of edges of CD(k, q), then e = Cl(v +*-<+i ). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g > 5 , g ^ 11,12.For g > 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 < g < 23 , g / 11, 12 , it improves on or ties existing bounds.
In this paper, we study $r$-uniform hypergraphs ${\cal H}$ without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for $r = 3$, we show that if ${\cal H}$ has $n$ vertices and a maximum number of edges, then $$|{\cal H}|={\textstyle 1\over6}n^{3/2} + o(n^{3/2}).$$ This also asymptotically determines the generalized Turán number $T_{3}(n,8,4)$. Some results are based on our bounds for the maximum size of Sidon-type sets in $\Bbb{Z}_{n}$.
We derive bounds for flu), the maximum number of edges in a graph on u vertices that contains neither three-cycles nor four-cycles. Also, w e give the exact value of flu) for all u up to 24 and constructive lower bounds for all u up to 200.
For any prime power q ≥ 3, we consider two infinite series of bipartite q-regular edge-transitive graphs of orders 2q 3 and 2q 5 which are induced subgraphs of regular generalized 4-gon and 6-gon, respectively. We compare these two series with two families of graphs, H 3 (p) and H 5 (p), p is a prime, constructed recently by Wenger ([26]), which are new examples of extremal graphs without 6and 10-cycles respectively. We prove that the first series contains the family H 3 (p) for q = p ≥ 3. Then we show that no member of the second family H 5 (p) is a subgraph of a generalized 6-gon. Then, for infinitely many values of q, we construct a new infinite series of bipartite q-regular edge-transitive graphs of order 2q 5 and girth 10. Finally, for any prime power q ≥ 3, we construct a new infinite series of bipartite q-regular edge-transitive graphs of order 2q 9 and girth g ≥ 14. Our constructions were motivated by some results on embeddings of Chevalley group geometries in the corresponding Lie algebras and a construction of a blow-up for an incident system and a graph. 10 9 v 1+ 1 9 ,
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